Topological Methods in Nonlinear Analysis

Realization of a graph as the Reeb graph of a Morse function on a manifold

Łukasz Patryk Michalak

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We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.

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Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 749-762.

First available in Project Euclid: 9 August 2018

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Michalak, Łukasz Patryk. Realization of a graph as the Reeb graph of a Morse function on a manifold. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749--762. doi:10.12775/TMNA.2018.029.

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